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Excellent Abstract Elementary Classes are tame
The assumption that an AEC is tame is a powerful assumption permitting
development of stability theory for AECs with the amalgamation property. Lately
several upward categoricity theorems were discovered where tameness replaces
strong set-theoretic assumptions.
We present in this article two sufficient conditions for tameness, both in
form of strong amalgamation properties that occur in nature. One of them was
used recently to prove that several Hrushovski classes are tame.
This is done by introducing the property of weak -uniqueness which
makes sense for all AECs (unlike Shelah's original property) and derive it from
the assumption that weak (\LS(\K),n)-uniqueness, (\LS(\K),n)-symmetry and
(\LS(\K),n)-existence properties hold for all . The conjunction of
these three properties we call \emph{excellence}, unlike \cite{Sh 87b} we do
not require the very strong (\LS(\K),n)-uniqueness, nor we assume that the
members of \K are atomic models of a countable first order theory. We also
work in a more general context than Shelah's good frames.Comment: 26 page
Short-time critical dynamics of the three-dimensional systems with long-range correlated disorder
Monte Carlo simulations of the short-time dynamic behavior are reported for
three-dimensional Ising and XY models with long-range correlated disorder at
criticality, in the case corresponding to linear defects. The static and
dynamic critical exponents are determined for systems starting separately from
ordered and disordered initial states. The obtained values of the exponents are
in a good agreement with results of the field-theoretic description of the
critical behavior of these models in the two-loop approximation and with our
results of Monte Carlo simulations of three-dimensional Ising model in
equilibrium state.Comment: 24 RevTeX pages, 12 figure
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